A generalized Charney—Stern theorem for semi-geostrophic dynamics

Abstract

A stability theorem for zonal flow through a channel bounded by zonally invariant topography is derived for semi-geostrophic (SG) dynamics. The stability theorem generalizes the quasi-geostrophic linear stability theorem of Charney and Stern. The SG stability theorem takes the form of a conservation law for a quadratic disturbance quantity whose sign-definiteness yields necessary and sufficient criteria for the basic flow to be stable. The disturbance quantity generalizes the linearized SG pseudomomentum wave activity for a rectangular channel derived by Kushner and Shepherd to a channel with more general topography. The conservation law is obtained by direct manipulation of the SG equations after they have been transformed to isentropic and geostrophic coordinates. The stability criteria depend on the basic state’s meridional potential-vorticity gradients in the interior, and on the topographic slope and the shear of the basic-state velocity at bounding material surfaces, but do not depend on the form of the small-amplitude disturbance.